We want to spend around 12 nights in Turkey from 21 May to 3 June. We have been to Istanbul so were thinking to skip it. We were really fascinated by the idea of south-eastern Anatolia as we love archaeology and dream to see Northern Mesopotamia. We are at a loss as to spend the rest of our time (A) further east and do Van, Kars and Trabzon (mainly to see Ani and the Sumela) or (B) drive from Gaziantep to Kayseri for Cappadocia (or/and fly to Izmir for Ephesus, Aphrodisia etc.). It will still be too cold to swim so not the time for a beach holiday probably.

We prefer seaside to mountains but have been all over the Med being Greek/Italians so are afraid the Mediterranean part could be less exciting and we hate mass tourism. At the same time, Van and Kars do not seem that exciting and we are knackered so would like nice hotels at some point.

I am thinking a compromise could be to do Cappadocia but stay in less touristic areas and possibly do the same for the Ephesus area.

Any suggestions?

I didn't make it as far east as you are considering, but I spent a chunk of time further west, including Cappadocia, Ephesus, Aphrodisia, and various places along or near the shore. I must admit that I think it might be a challenge to find "less touristic areas" in Cappadocia or near Ephesus -- even in 2010, there were MANY tourists in those locations. But I could be wrong!

It's old, but you might find some useful information in my trip report:
Some Observations on a Fantastic Journey through Western Turkey

Good luck with your decision!

Beach life is perfectly posssible in early June.
Take a bus from Cappadocia to Silifke and head then to the East (Narlikuyu, Kiskalesi, etc.) or - much better - to the West
(Yesilovacik, Kupala Beach Hotel Büyükeceli, Ulu Resort Hotel Yanisli, Aydincik, Anamur........) and fly home from Gazipasa.
There are also cheap flights from Kayseri, Diyarbakir, Urfa, Antep and Adana Cyprus.